Graph for the sequence - sequences and series, Mathematics

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Graph for the Sequence

First we wish to think about the term graphing a sequence. To graph the sequence {an} we plot the points {n, an} as n ranges over every possible values on a graph. For example, let's graph the sequence {n+1 / n2} n =1. The 1st few important points on the graph are,

 (1,2) , (2, ¾ ) , (3, 4/9) , (4, 5/16) , (5, 6/25), ....

After that the graph, for the first 30 terms of the sequence is as follow:

644_Graph for the Sequence - Sequences and Series 1.png

This graph leads us to a major idea regarding to the sequences.  Note: As n increases the sequence terms in our sequence, in this example, get closer and closer to zero.  After that we say that zero is the limit (or sometimes the limiting value) of the sequence and write,

1488_Graph for the Sequence - Sequences and Series 2.png

This note should look well known to you.  It is similar notation we used while we talked about the limit of a function.  Actually, if you remind, we said previous that we could think of sequences as functions in some way and thus this notation shouldn't be also surprising.


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